# Proof of special case of l’Hôpital’s rule | Differential Calculus | Khan Academy

What I want to go
over in this video is a special case
of L’Hopital’s Rule. And it’s a more
constrained version of the general case
we’ve been looking at. But it’s still very powerful
and very applicable. And the reason why we’re going
to go over this special case is because its proof is
fairly straightforward and will give you an intuition
for why L’Hopital’s Rule works at all. So the special case
of L’Hopital’s Rule is a situation where
f of a is equal to 0. f prime of a exists. g of a is equal to 0. g prime of a exists. If these constraints
are met, then the limit, as x approaches a of
f of x over g of x, is going to be equal to f
prime of a over g prime of a. So it’s very similar
to the general case. It’s little bit
more constrained. We’re assuming that
f prime of a exists. We’re not just
taking the limit now. We’re assuming f prime of a and
g prime of a actually exist. But notice if we substitute
a right over here we get 0/0. But that if the
derivatives exist we could just evaluate
the derivatives at a, and then we get the limit. So this is very close
to the general case of L’Hopital’s Rule. Now let’s actually prove it. And to prove it, we’re going
to start with the right hand and then show that if we use
the definition of derivatives, we get the left hand
right over here. So let me do that. So I’ll do it right over here. So f prime of a
is equal to what, by the definition
of derivatives? Well, we could view
that as the limit as x approaches a of f of x
minus f of a over x minus a. So this is literally just
a slope between two points. So like, if you have your
function f of x like this, this is the point a, f
of a right over here. This right over here
is the point x, f of x. This expression
right over here is the slope between
these two points. The change in our y value
is f of x minus f of a. The change in our f
value is x minus a. So this expression is just
the slope of this line. And we’re just taking
the– let me actually do that in a different
color– the line that connects these two points,
that’s the slope of it. I’ll do that in white. The slope of the line that
connects those two points. And we’re taking
the limit as x gets closer and closer
and closer to a. So this is just
another way of writing the definition of
the derivative. So that’s fine. Let’s do the same
thing for g prime of a. So f prime of a
over g prime of a, is going to be this
business which is in orange, f prime of a over g prime of a. Which we can write as
the limit as x approaches a of g of x minus g
of a over x minus a. Well, in the numerator,
we’re taking the limit as x approaches a, and
in the denominator, we’re taking the limit
as x approaches a. So we can just rewrite this. This we can rewrite as
the limit as x approaches a of all this
business in orange. f of x minus f of
a, over x minus a, over all the business in green. g of x minus g of a, all
of that over x minus a. Now, to simplify this, we
can multiply the numerator and the denominator by x minus
a to get rid of these x minus a’s. So let’s do that. Let’s multiply by x
minus a over x minus a. So the numerator, x minus a,
and we’re dividing by x minus a. Those cancel out. And then these two cancel out. And we’re left with this thing
over here is equal to the limit as x approaches a
of, in the numerator we have f of x minus f of a. And in the denominator, we
have g of x minus g of a. And I think you see
where this is going. What is f(a) equal to? Well, we assumed f
of a is equal to 0. That’s why we’re
using L’Hopital’s Rule from the get go. f of a is equal
to 0, g of a is equal to 0. f of a is equal to 0.
g of a is equal to 0. And this simplifies to the
limit as x approaches a of f prime of x, sorry of f of
x, we’ve got to be careful. Of f of x over g of x. So we just showed that if f of
a equals 0, g of a equals 0, and these two derivatives exist,
then the derivatives evaluated at a over each other are
going to be equal to the limit as x approaches a of
f of x over g of x. Or the limit as x
approaches a of f of x over g of x is going to
be equal to f prime of a over g prime of a. So fairly straightforward
proof for the special case– the special case, not
the more general case– of L’Hopital’s Rule.

### 21 thoughts on “Proof of special case of l’Hôpital’s rule | Differential Calculus | Khan Academy”

• February 16, 2013 at 12:23 am

Hi Sal. You did a proof of the Fundamenta theorem of Calculus, but relied there on mean value theorem, which there's no video with proof of. Could you please make a video on that? thank you.

• February 16, 2013 at 1:36 am

huh

• February 16, 2013 at 7:12 am

good …

• February 16, 2013 at 9:15 am

look, i never said that i didn't have the intuition. The intuition is there, and it's pretty strong. But having the intuition is one thing, and having a rigorous proof is another. When you learn mathematics, you should feel like a judge. And a judge seldom makes conclusions based on intuition, rather, he would look for a convincing argument. What intuition does is points out a way to go to find the argument.

• February 16, 2013 at 3:27 pm

Slight nitpick: I think you need to say that g'(a) must be non-zero.

• February 17, 2013 at 6:19 pm

I actually like L'hospital's Rule, It does simplify finding the limit

• April 25, 2013 at 12:35 pm

@1:04 gr8 choice of colors , sal. 🙂
the colors are exactly in according to the Indian flag !! 😀

• May 30, 2013 at 11:40 am

What does he mean by saying with respect to x and y or xy

• June 25, 2013 at 2:24 am

Wow.
I was surfing for boobs and ended up here in the weird part of youtube again.
I would go to bed but this is fucking spectacular! So…
If F(a) is the left boob and G(a) is the right boob then F'(a) / G'(a) = naked woman lying on her right side.
Mind blown. Time for bed, y'all. I'm going to be the numerator and my fucking mattress is going to be my denominator. It's your denominator too, niggaz! G'night.

• September 30, 2013 at 9:43 am

I looooooove you

• March 23, 2014 at 4:54 pm

Hello. Great video. Could you please tell me where I can find this special case of l'Hopital rule written formally. Maybe on some official calculus paper or something like that?

• December 6, 2016 at 1:19 pm

Sir can you make a video on explaining l hopitals rule on sums like"log(log(sin(tan(x^2-2x)^3", this is just an example, but sums like this !?!?

• January 30, 2017 at 11:39 pm

Confused ! So I assume that one has to prove the limit exists before one can take it, which begs the question…….how can this be done without actually taking the limit?  This appears to be a contradiction in terms, or at least ambiguous. Any clarification would be appreciated and thanks for great insightful videos.

• June 4, 2017 at 12:44 pm

what is the proof behind infinity/infinity?

• January 31, 2018 at 12:21 pm

Well .. some indefinities here and there but it's ok , thats not a video for mathematicians . Great job for sharing knowledge with the world !!!

• January 2, 2019 at 10:51 pm

Sal. Khan, I wish u were a professor at my university :/

• January 5, 2019 at 12:30 pm

Hopital should be afraid from you!!! 🙂

• November 14, 2019 at 8:16 pm

Why do we use the mean value theorem rather than the definition of a derivative?

• December 2, 2019 at 12:34 am